# Automorphism-Induced Non-Canonicity in Top-k Explanations of Graph Neural Networks

arXiv: https://arxiv.org/abs/2607.26344 · HTML: https://mlsanity.com/paper/2607.26344/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 27 references that are in the map.
- First posted on arXiv 28 Jul 2026.

## Builds on (most cited first, 18 of 18)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Axiomatic Attribution for Deep Networks](https://mlsanity.com/paper/1703.01365/) (2017)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [MoleculeNet: A Benchmark for Molecular Machine Learning](https://mlsanity.com/paper/1703.00564/) (2017)
- [Strategies for Pre-training Graph Neural Networks](https://mlsanity.com/paper/1905.12265/) (2019)
- [Weisfeiler and Leman Go Neural: Higher-order Graph Neural Networks](https://mlsanity.com/paper/1810.02244/) (2018)
- [GNNExplainer: Generating Explanations for Graph Neural Networks](https://mlsanity.com/paper/1903.03894/) (2019)
- [TUDataset: A collection of benchmark datasets for learning with graphs](https://mlsanity.com/paper/2007.08663/) (2020)
- [Parameterized Explainer for Graph Neural Network](https://mlsanity.com/paper/2011.04573/) (2020)
- [GraphSVX: Shapley Value Explanations for Graph Neural Networks](https://mlsanity.com/paper/2104.10482/) (2021)
- [Finding Symmetry Breaking Order Parameters with Euclidean Neural Networks](https://mlsanity.com/paper/2007.02005/) (2020)
- [Equivariance Everywhere All At Once: A Recipe for Graph Foundation Models](https://mlsanity.com/paper/2506.14291/) (2025)
- [Reconsidering Faithfulness in Regular, Self-Explainable and Domain Invariant GNNs](https://mlsanity.com/paper/2406.15156/) (2024)
- [Evaluating the Robustness of Interpretability Methods through Explanation Invariance and Equivariance](https://mlsanity.com/paper/2304.06715/) (2023)
- [GNNX-BENCH: Unravelling the Utility of Perturbation-based GNN Explainers through In-depth Benchmarking](https://mlsanity.com/paper/2310.01794/) (2023)
- [Explaining Graph Neural Networks via Structure-aware Interaction Index](https://mlsanity.com/paper/2405.14352/) (2024)
- [Robust Ante-hoc Graph Explainer using Bilevel Optimization](https://mlsanity.com/paper/2305.15745/) (2023)

## Abstract (arXiv)

A gradient-based GNN explainer given a molecule with two chemically equivalent nitro groups assigns them attribution scores that are equal to the last bit. It cannot do otherwise: message passing is exactly permutation equivariant, so any automorphism of the input leaves every attribution invariant. Yet the standard report, the top-k edges, names one of the two, and which one is settled by the order of an array. We show this is a structural obstruction rather than an implementation slip. When no minimal valid explanation is fixed by the input's automorphism group, no rule can be single-valued, minimal and symmetry-respecting at once. For the exact-k reports used in practice we give a parameter-free criterion, mechanised in Lean 4 with no axiom dependencies, that decides from the graph alone whether every score-optimal report of that size must split an orbit. Across 21298 instance-budget decisions the criterion agrees with a mechanical model-equivalence check without exception, and no severing case we found admitted a neutral alternative. The obstruction is common. Nontrivial automorphisms occur in 93.4% of Mutagenicity, the dataset the seminal explainability papers use, so the measure-zero dismissal of symmetric inputs, sound on the continuous domains it was made for, collapses here. At the sparsity budget those papers report, 24.0% of molecules with two interchangeable nitro groups (6 of 25) surface exactly one of them, every one arbitrary under mechanical verification. A model's blindness also manufactures symmetry: every MUTAG molecule contains atoms chemistry separates and the network provably cannot, and a matched control shows the resolution is set by what the model reads rather than how it is parameterised. Reporting orbits removes the arbitrariness at 0.11 ms and 0.43 extra edges per graph.